Guiding Center

By solving the trajectories of particles, we can calculate the actual guiding center orbits by following the definition. This is supported directly via get_gc. TestParticle.jl also provides several methods to trace the guiding center, serving different purposes.

Diagnostic Methods (Reference-based)

These methods require a full particle trajectory solution to calculate drifts. They are useful for analyzing which drift components dominate or checking the validity of drift approximations.

• trace_gc_drifts!: Calculates the guiding center trajectory using standard analytical drift formulas (ExB, Gradient-B, Curvature drifts). It uses the parallel and perpendicular velocities from the full particle simulation.

• trace_gc_exb!: Calculates the trajectory considering only the $\mathbf{E}\times \mathbf{B}$ drift and parallel motion. Useful for identifying ExB dominance.

• trace_gc_flr!: Includes Finite Larmor Radius (FLR) corrections to the ExB drift, suitable for nonuniform electric fields.

Self-consistent Solvers (GCA)

trace_gc! solves the 1st order Guiding Center Approximation (GCA) equations. This approximation holds when the characteristic scale of field variations $L$ (often represented by the curvature radius) is much larger than the Larmor radius $\rho$ ($L\gg \rho$).

State Variables: The solver evolves the 4D state $[\mathbf{R},v_{\parallel }]$:

• $\mathbf{R}$: The spatial position of the guiding center.

• $v_{\parallel }$: The velocity parallel to the magnetic field.

Parameters:

• $\mu =p_{\perp }^2/(2mB)$: The magnetic moment (adiabatic invariant). It is calculated from initial conditions and treated as a constant parameter.

Effective Fields:

To incorporate geometric drifts (curvature) and mirror forces naturally, we define the effective magnetic and electric fields:

$$\begin{array}{rl}{\mathbf{B}}^{\ast }& =\mathbf{B}+\frac{m{v}_{\parallel }}{q}\mathrm{\nabla }\times \mathbf{b}\\ {\mathbf{E}}^{\ast }& =\mathbf{E}-\frac{1}{q}\mathrm{\nabla }(\mu B)\end{array}$$

where $\mathbf{b}=\mathbf{B}/B$ is the unit vector along the magnetic field.

Equations of Motion:

The time evolution of the guiding center is governed by:

$$\begin{array}{rl}\dot{\mathbf{R}}& =\frac{1}{B_{\parallel }^{\ast }}[v_{\parallel }{\mathbf{B}}^{\ast }+{\mathbf{E}}^{\ast }\times \mathbf{b}]\\ {\dot{v}}_{\parallel }& =\frac{q}{m{B}_{\parallel }^{\ast }}{\mathbf{B}}^{\ast }\cdot {\mathbf{E}}^{\ast }\end{array}$$

where $B_{\parallel }^{\ast }=\mathbf{b}\cdot {\mathbf{B}}^{\ast }$.

• Term 1 of $\dot{\mathbf{R}}$ ($\propto {\mathbf{B}}^{\ast }$) represents motion along the field line plus curvature drift.

• Term 2 of $\dot{\mathbf{R}}$ ($\propto {\mathbf{E}}^{\ast }\times \mathbf{b}$) represents the $\mathbf{E}\times \mathbf{B}$ drift and the $\mathrm{\nabla }B$ drift.

In theoretical treatments, individual drift terms are often derived separately (see below). By solving the GCA equations, all these drifts are captured self-consistently to first order.

Summary of guiding center drifts

General force:

$${\mathbf{v}}_f=\frac{1}{q}\frac{\mathbf{F}\times \mathbf{B}}{B^2}$$

Electric field:

$${\mathbf{v}}_E=\frac{\mathbf{E}\times \mathbf{B}}{B^2}$$

Gravitational field:

$${\mathbf{v}}_g=\frac{m}{q}\frac{\mathbf{g}\times \mathbf{B}}{B^2}$$

Nonuniform electric field:

$${\mathbf{v}}_E=(1+\frac{1}{4}{r}_L^2{\mathrm{\nabla }}^2)\frac{\mathbf{E}\times \mathbf{B}}{B^2}$$

Nonuniform magnetic field:

Grad-B:

$$\begin{array}{rl}{\mathbf{v}}_{\mathrm{\nabla }B}& =\pm \frac{1}{2}{v}_{\perp }{r}_L\frac{\mathbf{B}\times \mathrm{\nabla }B}{B^2}\\ & =\frac{m{v}_{\perp }^2}{2q{B}^3}\mathbf{B}\times \mathrm{\nabla }B\end{array}$$

Curvature drift:

$$\begin{array}{rl}{\mathbf{v}}_c& =\frac{m{v}_{\parallel }^2}{q}\frac{{\mathbf{R}}_c\times \mathbf{B}}{R_c^2{B}^2}\\ & =\frac{m{v}_{\parallel }^2}{q{B}^4}\mathbf{B}\times [\mathbf{B}\cdot \mathrm{\nabla }\mathbf{B}]\\ & =\frac{m{v}_{\parallel }^2}{q{B}^3}\mathbf{B}\times \mathrm{\nabla }B\text{(current-free)}\end{array}$$

Curved vacuum field:

$$\begin{array}{rl}{\mathbf{v}}_c+{\mathbf{v}}_{\mathrm{\nabla }B}& =\frac{m}{q}(v_{\parallel }^2+\frac{1}{2}{v}_{\perp }^2)\frac{{\mathbf{R}}_c\times \mathbf{B}}{R_c^2{B}^2}\\ & =\frac{m}{q}\frac{\mathbf{B}\times \mathrm{\nabla }B}{B^3}(v_{\parallel }^2+\frac{1}{2}{v}_{\perp }^2)\end{array}$$

Polarization drift:^[1]

$${\mathbf{v}}_p=\pm \frac{1}{{\omega }_{c}B}\frac{d\mathbf{E}}{dt}$$

Adiabatic Invariants

See more thorough notes on the adiabatic invariants.

---

1. In common test particle models, we assume static EM fields, so the polarization drift as well as adiabatic heating is not present. However, it is easily achieveable here in this package. ↩︎